1. Technical Field
The present invention relates to data processing in general, and in particular to an apparatus for solving differential equations. Still more particularly, the present invention relates to an apparatus for solving very-large scale systems of time-continuous differential equations.
2. Description of Related Art
Generally, analog computers can be set up to solve very-large scale systems of time-continuous differential equations. However, difficulties may arise due to a number of factors. For example, magnitude scaling problems arise when analog components exceed the limits imposed by supply voltages, and time scaling problems may occur from time constants associated with analog integrators. In addition, accuracy problems can be caused by factors such as components not being of desired values, circuit analysis approximations, and intrinsic electronic noise generated by resistors (thermal noise), direct currents (shot noise), active devices (flicker and popcorn noise), etc. Hence, although analog computers can be a powerful tool for solving very-large scale systems of differential equations, their power is severely restrained by the above-mentioned problems.
On the other hand, time-continuous differential equations may be simulated and numerically solved by using digital computers with an appropriate numerical method. With derivatives approximated by finite differences, a system of time-continuous differential equations can be converted into an equivalent system of time-discrete difference equations. However, solving a dynamic system using difference equations may lead to numerical instability and accuracy problems, which include stiff system issues, function smoothness difficulties, rounding and truncation errors, and error buildup. Thus, the usage of digital computers to solve time-continuous differential equations also suffer from various limitations.
Consequently, it would be desirable to provide an improved apparatus for solving very-large scale systems of time-continuous differential equations.